Iwaniec–Martin conjecture for the Donaldson–Sullivan signature operator

Let $S$ be the Donaldson–Sullivan signature operator on differential forms over $\mathbb{R}^n$, defined by

$$S\omega=(\delta\delta^*-\delta^*\delta)(-\Delta)^{-1}\omega.$$

For $1<p<\infty$, let $p^*=\max\{p,p/(p-1)\}$. Iwaniec–Martin conjecture. For every $n\geq 2$,

$$\|S\|_p=p^*-1.$$

In dimension two, on one-forms, $S$ reduces to the Beurling–Ahlfors operator up to sign. The source reports the lower bound $p^*-1$ and a nonsharp upper bound, with no resolution of the conjecture.